chapter 3 - Bentham Science

84 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool, 2011, 84–106
Computational Problem Solving in Context:
From Arithmetic Sequences to Polygonal-like Numbers
Sergei Abramovich
Department of Curriculum and Instruction
State University of New York at Potsdam, Potsdam, New York, USA
CHAPTER 5
Address correspondence to: Prof. Sergei Abramovich, Department of Curriculum and Instruction, State
University of New York, 210 Satterlee Hall, Potsdam, New York 13676, USA; Tel: (+1) 315-267-2541;
E-mail: abramovs@potsdam.edu
Abstract: The learning of mathematical concepts can be aided by the use of technology
and graphical visualization. The use of modern tools can greatly enhance the
learning experience by encouraging inquiry and deepening one’s understanding. In
this chapter triangular, square, and polygonal-like numbers are explored in context
using spreadsheets. The context provides the element of relevance that learners need
in order to appreciate concepts that might otherwise seem abstract in nature.
Keywords: contextual problem solving, arithmetic sequences, polygon-like numbers.
5.1 Introduction
A number sequence in which the difference between any two consecutive terms is a constant is
called an arithmetic sequence. The simplest example of such a sequence is the sequence of counting
numbers 1, 2, 3, 4, ..., where the difference between any two consecutive terms is one. Similarly,
one can consider the sequence of consecutive odd (1, 3, 5, 7, ...) or even (2, 4, 6, 8, ...) numbers
as arithmetic sequences with the difference two.
Many problems in mathematics lead to the summation of consecutive counting, odd, or even
numbers. Finding such sums can be extended to the summation of other arithmetic sequences.
Partial sums of arithmetic sequences (e.g., 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, and so
on) form new number sequences known as polygonal numbers [1]. In particular, these numbers,
having strong connection to geometry, include triangular (1, 3, 6, 10, ...) and square (1, 4, 9, 16,
...) numbers which can already be found in the elementary school curriculum as links between
different mathematical concepts (e.g., [2]).
An arithmetic sequence is one of the most basic concepts of algebra and number theory. The
study of arithmetic sequences and their partial sums lends itself to the joint use of context and
technology. In particular, the use of a spreadsheet in teaching algebra and number theory topics
Mark Lau and Stephen Sugden (Eds)
All rights reserved – c○2011 Bentham Science Publishers Ltd.